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The Cut-Elimination Theorem for Differential Nets with Promotion [chapter]

Michele Pagani
2009 Lecture Notes in Computer Science  
We extend this analysis to exponential boxes and prove the Cut-Elimination Theorem for the whole DiLL: every differential net that is sequentializable can be reduced to a cut-free net.  ...  The authors have examined the cut-elimination of the promotion-free fragment of DiLL by means of a proofnet-like calculus: differential interaction nets.  ...  The Cut-Elimination Theorem We prove our main result: switching acyclic differential nets enjoy the cutelimination (Theorem 1).  ... 
doi:10.1007/978-3-642-02273-9_17 fatcat:5tep524tprgohnxvsh6jfktn2y

Visible acyclic differential nets, Part I: Semantics

Michele Pagani
2012 Annals of Pure and Applied Logic  
Visible acyclicity discloses a new kind of correctness for the promotion rule of linear logic, which goes beyond sequent calculus correctness.  ...  (Michele Pagani) 1 Precisely wires also represent the identity rules of axiom and cut, following the spirit of Lafont's interaction nets [2] .  ...  Acknowledgements I am especially grateful to Thomas Ehrhard for his several hints which let me understand finiteness spaces and differential linear logic.  ... 
doi:10.1016/j.apal.2011.09.001 fatcat:gkqlo42mcvcopmpfiflbbyvjli

The conservation theorem for differential nets

MICHELE PAGANI, PAOLO TRANQUILLI
2015 Mathematical Structures in Computer Science  
We prove the conservation theorem for differential netsthe graph-theoretical syntax of the differential extension of Linear Logic (Ehrhard and Regnier's DiLL).  ...  The conservation theorem states that the property of having infinite reductions (here infinite chains of cut elimination steps) is preserved by non-erasing steps.  ...  In this paper we prove an analogous theorem for differential nets -the graph-theoretical syntax of the differential extension of Linear Logic (DiLL [ER06] ).  ... 
doi:10.1017/s0960129515000456 fatcat:o44phpxvczflboi3jlcra6xg6u

An Application of Parallel Cut Elimination in Unit-Free Multiplicative Linear Logic to the Taylor Expansion of Proof Nets

Jules Chouquet, Lionel Vaux Auclair, Michael Wagner
2018 Annual Conference for Computer Science Logic  
In order to simulate one cut elimination step in MELL, it is necessary to reduce an arbitrary number of cuts in the differential nets of its Taylor expansion.  ...  It turns out our results apply to differential nets, because their cut elimination is essentially multiplicative.  ...  The target of Taylor expansion is then in promotion-free differential nets [13] , which we call resource nets in the following, by analogy with resource λ-calculus: these form the multilinear fragment  ... 
doi:10.4230/lipics.csl.2018.15 dblp:conf/csl/ChouquetA18 fatcat:6pchgimstjeffcndp2rnsjp7pu

An application of parallel cut elimination in multiplicative linear logic to the Taylor expansion of proof nets [article]

Jules Chouquet, Lionel Vaux Auclair
2021 arXiv   pre-print
In order to simulate one cut elimination step in MELL, it is necessary to reduce an arbitrary number of cuts in the differential nets of its Taylor expansion.  ...  It turns out our results apply to differential nets, because their cut elimination is essentially multiplicative.  ...  actually taken on the occasion of the very first workshop of the network at Roma Tre in December 2015, dedicated to New trends in linear logic proof-nets; and successive versions were presented and discussed  ... 
arXiv:1902.05193v6 fatcat:hz55exurljckzly4nhjmimw5ea

Confluence of Pure Differential Nets with Promotion [chapter]

Paolo Tranquilli
2009 Lecture Notes in Computer Science  
We study the confluence of Ehrhard and Regnier's differential nets with exponential promotion, in a pure setting.  ...  This result generalizes to linear logic regular proof nets, where the same notion of equivalence was already studied in the literature, but only with respect to the problem of normalization in a typed  ...  One of these, central to our work, is confluence of cut elimination, i.e. the independence of the result of the cut elimination procedure with respect to the actual cuts one decides to reduce.  ... 
doi:10.1007/978-3-642-04027-6_36 fatcat:yt4hntn7w5esjcd4oequg7ee24

An application of parallel cut elimination in multiplicative linear logic to the Taylor expansion of proof nets

Jules Chouquet, Lionel Vaux Auclair
2021 Logical Methods in Computer Science  
In order to simulate one cut elimination step in MELL, it is necessary to reduce an arbitrary number of cuts in the differential nets of its Taylor expansion.  ...  It turns out our results apply to differential nets, because their cut elimination is essentially multiplicative.  ...  actually taken on the occasion of the very first workshop of the network at Roma Tre in December 2015, dedicated to New trends in linear logic proof-nets; and successive versions were presented and discussed  ... 
doi:10.46298/lmcs-17(4:22)2021 fatcat:7d5vr44auvfmhmb6zonjqfgwd4

Preface

LORENZO TORTORA DE FALCO
2016 Mathematical Structures in Computer Science  
of the proof-net/term α is itself a (differential) proof-net/term and an approximation of α.  ...  and developed much later, like differentiation and Taylor expansion for proofs.  ...  Preface 992 proof of the completeness theorem for first-order logic.  ... 
doi:10.1017/s096012951600044x fatcat:rdlnum3ddvetxh3p5arwzlp2gy

Linear Logic and Strong Normalization

Beniamino Accattoli, Marc Herbstritt
2013 International Conference on Rewriting Techniques and Applications  
Strong normalization for linear logic requires elaborated rewriting techniques. In this paper we give a new presentation of MELL proof nets, without any commutative cut-elimination rule.  ...  Moreover, it is an axiomatic proof, as more generally it holds for every set of rewriting rules satisfying three very natural requirements with respect to substitution, commutation with promotion, full  ...  Acknowledgements To Delia Kesner, who taught me the art of strong normalization proofs, and with whom I had uncountable discussions on commutative rules, normalization, and proof nets.  ... 
doi:10.4230/lipics.rta.2013.39 dblp:conf/rta/Accattoli13 fatcat:niytow7rsbhuxabwthipfbwlti

Interpreting a finitary pi-calculus in differential interaction nets

Thomas Ehrhard, Olivier Laurent
2010 Information and Computation  
To simplify the presentation, these nets use only a restricted form of the promotion rule of linear logic, which is sufficient for interpreting a replication-free version of the π-calculus, as well as  ...  This shows that differential interaction nets are sufficiently expressive for representing concurrency and mobility, as formalized by the pi-calculus.  ...  They correspond to the cut elimination rules of the differential linear logic of Section 2.1.  ... 
doi:10.1016/j.ic.2009.06.005 fatcat:4fwmomneuzgwjdihbbqvqenzmi

Logic and linear algebra: an introduction [article]

Daniel Murfet
2017 arXiv   pre-print
We give an introduction to logic tailored for algebraists, explaining how proofs in linear logic can be viewed as algorithms for constructing morphisms in symmetric closed monoidal categories with additional  ...  The interesting part of this vector space semantics is based on the cofree cocommutative coalgebra of Sweedler.  ...  The main theorem is that the relation is weakly normalising, which means: Theorem 6.7 (Cut-elimination).  ... 
arXiv:1407.2650v3 fatcat:hjiw7vripnecdk5352eznu7nwe

Proof Nets and the Linear Substitution Calculus [article]

Beniamino Accattoli
2018 arXiv   pre-print
Nonetheless, two different terms with sharing can still have the same proof nets representation---a further result is the characterisation of the equality induced by proof nets over terms with sharing.  ...  Here we show that the linear substitution calculus, a simple refinement of the λ-calculus with sharing, is isomorphic to proof nets at the operational level.  ...  To the reviewers, for useful comments. This work has been partially funded by the ANR JCJC grant COCA HOLA (ANR-16-CE40-004-01).  ... 
arXiv:1808.03395v1 fatcat:g4x23aiusvfcxi5dgttxdqoc7q

Proof Nets and the Linear Substitution Calculus [chapter]

Beniamino Accattoli
2018 Lecture Notes in Computer Science  
Nonetheless, two different terms with sharing can still have the same proof nets representation-a further result is the characterisation of the equality induced by proof nets over terms with sharing.  ...  Here we show that the linear substitution calculus, a simple refinement of the λ-calculus with sharing, is isomorphic to proof nets at the operational level.  ...  To the reviewers, for useful comments. This work has been partially funded by the ANR JCJC grant COCA HOLA (ANR-16-CE40-004-01).  ... 
doi:10.1007/978-3-030-02508-3_3 fatcat:37gfbyblnfgkxp2o4c2s7eopry

Geometry of resource interaction and Taylor–Ehrhard–Regnier expansion: a minimalist approach

MARCO SOLIERI
2016 Mathematical Structures in Computer Science  
The resource λ-calculus is a variation of the λ-calculus where arguments are superpositions of terms and must be linearly used; hence, it is a model for linear and non-deterministic programming languages  ...  Lastly, we also provide an expanded counterpart of the execution formula, which computes paths as series of objects of GoRI; thus, exchanging determinism and conciseness for linearity and simplicity.  ...  Path dynamics With the notions just introduced we now define the property of path persistence, that intuitively means "surviving cut-elimination".  ... 
doi:10.1017/s0960129516000311 fatcat:tfth2yb52rhxxaioawcmgaigpa

Proofs as Executions [chapter]

Emmanuel Beffara, Virgile Mogbil
2012 Lecture Notes in Computer Science  
A type system based on linear logic is used, in which a given process has many different types, each typing corresponding to a particular way of interacting with its environment and cut elimination corresponds  ...  to executing the process in a given interaction scenario.  ...  Section 4 presents the proof nets for the logic of schedulings and its cut-elimination.  ... 
doi:10.1007/978-3-642-33475-7_20 fatcat:oqbzzkfswfawxagsxkvkqscage
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